MATHEMATICS

I. I. IBRAGIMOV

ON ESTIMATING THE NORM OF A LINEAR OPERATOR IN A CLASS OF ENTIRE FUNCTIONS OF FINITE DEGREE

(Presented by Academician V. I. Smirnov on May 6, 1963)

1. Let \(W_{\nu_1,\ldots,\nu_n}^{(p)}\) \((p \geqslant 1)\) denote the class of entire functions \(g(z_1,\ldots,z_n)\) of degree \(\leqslant (\nu_1,\ldots,\nu_n)\) and satisfying the condition

\[ \bigl(\|g\|_{p}^{(n)}\bigr)^p = \int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty} |g(x_1,\ldots,x_n)|^p\,dx_1\cdots dx_n < +\infty . \]

Let, further, \(\mathfrak M\) be the set of linear operators \(T\) possessing the following properties:

\(1^\circ.\) \(T\) is defined on the set \(W_{\nu_1,\ldots,\nu_n}^{(p)}\), and its norm
\(\|T[g]\|_{p}^{(n)}\) is invariant with respect to any real shift in each argument.

\(2^\circ.\) There exists a constant \(A(\nu_1,\ldots,\nu_n)\) such that

\[ \|T[g]\|_{\infty}^{(n)} \leqslant A(\nu_1,\ldots,\nu_n)\,\|g\|_{p}^{(n)}, \]

where \(\|f\|_{\infty}^{(n)}\) denotes the norm of the function \(f(x_1,\ldots,x_n)\) in the metric of the space \(C^{(n)}(-\infty,\infty)\).

The first problem of the present note* is to establish, in the form of an inequality, a dependence between the various norms of different linear operators \(T\) and \(S\) from the set \(\mathfrak M\) in the class of entire functions \(W_{\nu_1,\ldots,\nu_n}^{(p)}\) \((p \geqslant 1)\). The solution of this problem is based on the fact that for any entire function \(g(z)=g(x+iy)\) from the class \(W_{\nu}^{(p)}\) \((p>1)\), for any \(z=x+iy\), the identity holds (see (1), p. 59)

\[ g(x+iy)=\frac{1}{\pi}\int_{-\infty}^{\infty} g(t-x)\,\frac{\sin \nu(t+iy)}{t+iy}\,dt . \tag{1} \]

By successive application of identity (1) to the function \(g(z_1,\ldots,z_n)\) with respect to each argument, we find that for an entire function \(g(z_1,\ldots,z_n)\) from the class \(W_{\nu_1,\ldots,\nu_n}^{(p)}\) \((p \geqslant 1)\) we have:

\[ g(x_1+iy_1,\ldots,x_n+iy_n)= \]

\[ = \left(\frac{1}{\pi}\right)^n \int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty} f(t_1-x_1,\ldots,t_n-x_n) \prod_{k=1}^{n} \frac{\sin \nu_k(t_k+iy_k)}{t_k+iy_k} \,dt_1\cdots dt_n . \tag{2} \]

Hence, when \(y_1=y_2=\cdots=y_n=0\), it follows that for an entire function
\(g(z_1,\ldots,z_n)\in W_{\nu_1,\ldots,\nu_n}^{(p)}\) \((p \geqslant 1)\), for any real \(x_1,\ldots,x_n\), the identity holds

\[ g(x_1,\ldots,x_n)= \]

\[ = \left(\frac{1}{\pi}\right)^n \int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty} g(t_1-x_1,\ldots,t_n-x_n) \prod_{k=1}^{n} \frac{\sin \nu_k t_k}{t_k} \,dt_1\cdots dt_k . \tag{3} \]

* The results of the present note were reported at the International Congress of Mathematicians in August 1962 in Stockholm and at the Second All-Union Conference on Constructive Function Theory in October 1962 in Baku.

Moreover, we use the fact that if \(g(z_1,\ldots,z_n)\in W^{(p)}_{\nu_1,\ldots,\nu_n}\) \((p\geqslant 1)\) and \(\varphi(x_1,\ldots,x_n)\in \mathscr{L}^{(n)}_{p'}\left(\dfrac1p+\dfrac1{p'}=1\right)\), then

\[ F(z_1,\ldots,z_n)=\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty} g(z_1+t_1,\ldots,z_n+t_n)\varphi(t_1,\ldots,t_n)\,dt_1\cdots dt_n \tag{4} \]

is an entire function of the class \(B_{\nu_1,\ldots,\nu_n}\) (see \((1)\), p. 38), and the function \(\varphi(t_1,\ldots,t_n)\) can be chosen so that the equality

\[ F(0,0,\ldots,0)=\|g\|^{(n)}_p \tag{5} \]

holds.

This leads to the following assertions:

Theorem 1. If \(g(z_1,\ldots,z_n)\) is an entire function of the class \(W^{(p)}_{\nu_1,\ldots,\nu_n}\) \((p\geqslant 1)\); \(T,S\) are linear operators from the set \(\mathfrak{M}\), then from the validity of the inequality*

\[ \|T[g]\|^{(n)}_\infty \leqslant A(\nu_1,\ldots,\nu_n)\|g\|^{(n)}_\infty \]

it follows that

\[ \|T[g]\|^{(n)}_p \leqslant A(\nu_1,\ldots,\nu_n)\|g\|^{(n)}_p \]

for every \(p\geqslant 1\). Moreover, from the validity of the inequality

\[ \|T[g]\|^{(n)}_\infty \leqslant \lambda(\nu_1,\ldots,\nu_n)\|S[g]\|^{(n)}_\infty \]

it follows that

\[ \|T[g]\|^{(n)}_p \leqslant \lambda(\nu_1,\ldots,\nu_n)\|S[g]\|^{(n)}_p \]

for every \(p\geqslant 1\).

In particular, if \(T\) is the differentiation operator, then from the validity of the classical inequality of S. N. Bernstein

\[ \left|\frac{\partial g(x_1,\ldots,x_n)}{\partial x_k}\right|\leqslant \nu_k\|g\|^{(n)}_\infty \]

there follows the Bernstein–Nikol’skii inequality \((3)\)

\[ \left\|\frac{\partial g(x_1,\ldots,x_n)}{\partial x_k}\right\|^{(n)}_p \leqslant \nu_k\|g\|^{(n)}_p \]

for every \(p\geqslant 1\).

Theorem 2. Let \(T,S\) be linear operators from the set \(\mathfrak{M}\), and let \(g(z_1,\ldots,z_n)\) be an entire function of the class \(W_{\nu_1,\ldots,\nu_n}\) \((p\geqslant 1)\). Then from the validity of the inequality

\[ \|T[g]\|^{(n)}_\infty \leqslant \lambda(\nu_1,\ldots,\nu_n)\|S[g]\|^{(n)}_\infty \]

it follows that

\[ \|T[g]\|^{(n)}_\infty \leqslant \prod_{k=1}^{n}\left(\frac{\nu_k}{\pi}\right)^{1/p} \lambda(\nu_1,\ldots,\nu_n)\|S[g]\|^{(n)}_p \]

for \(1\leqslant p\leqslant 2\), and, moreover,

\[ \|T[g]\|^{(n)}_q \leqslant \left(\prod_{k=1}^{n}\frac{\nu_k}{\pi}\right)^{1/p-1/q} \lambda(\nu_1,\ldots,\nu_n)\|S[g]\|^{(n)}_p \]

for \(1\leqslant p<q\leqslant +\infty\).

Theorem 3. If \(g(z_1,\ldots,z_n)\) is an entire function of the class \(W^{(p)}_{\nu_1,\ldots,\nu_n}\) \((p\geqslant 1)\), \(T\) is a linear operator from the set \(\mathfrak{M}\), \(1\leqslant p\leqslant 2\),

* Theorem 1 in the one-dimensional case was proved by another method in work \((2)\).

\(1 \leqslant p < q \leqslant +\infty\), then

\[ \|T[g]\|_{q}^{(n)} \leqslant \left[\left(\frac{1}{\pi}\right)^n B_q^{n/q} \left(\prod_{k=1}^{n} \nu_k\right)^{1/p}\right]^{1-p/q} \|T[g]\|_{p}^{(n)}, \]

where

\[ B_{\infty}=1,\qquad B_q=\int_{-\infty}^{\infty}\left|\frac{\sin u}{u}\right|^q\,du. \]

Hence, in the case when \(T\) is the identity transformation, the inequality

\[ \|g\|_{q}^{(n)} \leqslant \prod_{k=1}^{n}\left(\frac{s\nu_k}{\pi}\right)^{1/p-1/q} \|g\|_{p}^{(n)} \]

holds for all \(p\) and \(q\) satisfying the condition \(1 \leqslant p<q\leqslant+\infty\), where \(s=[[{-p/2}]]\) is the least integer not less than \(p/2\). The last inequality is a refinement of S. M. Nikol’skii’s inequality \((^3)\), which was generalized and sharpened in works \((^{4-6})\) for more general differential operators.

II. Let \(\varphi(x_1,\ldots,x_n)\geqslant 1\) be a continuous function in \(n\)-dimensional Euclidean space \((R_n)\), let \(p\geqslant 1\) be any number, and let \(\Lambda_{P,\varphi}\) be the class of functions \(f(x_1,\ldots,x_n)\) possessing the property

\[ \|f\|_{P,\varphi}^{(n)} = \left\|\ldots \left\{ \left\|\ldots \left(\left\|\frac{f}{\varphi}\right\|_{p_1}\right) \ldots\right\|_{p_k} \right\} \ldots\right\|_{p_n} <+\infty, \]

where \(P=(p_1,p_2,\ldots,p_n)\) and \(p_1,p_2,\ldots,p_n\) are various numbers not less than unity. In particular, for \(n=3\) the norm \(\|f\|_{P,\varphi}^{(3)}\) has the form:

\[ \|f\|_{P,\varphi}^{(3)} = \left\{ \int_{-\infty}^{\infty} \left[ \int_{-\infty}^{\infty} \left( \int_{-\infty}^{\infty} \left| \frac{f(x_1,x_2,x_3)}{\varphi(x_1,x_2,x_3)} \right|^{p_1} dx_1 \right)^{p_2/p_1} dx_2 \right]^{p_3/p_2} dx_3 \right\}^{1/p_3} <+\infty. \]

Obviously, the class \(\Lambda_{P,\varphi}^{(n)}\), called the generalized Lebesgue class, coincides with the ordinary Lebesgue class \(\mathcal L_p^{(n)}(-\infty,\infty)\) when

\[ \varphi(x_1,\ldots,x_n)\equiv 1,\qquad p_1=\cdots=p_n=p. \]

Further, let \(W_{\nu_1,\ldots,\nu_n}^{(P,\varphi)}\) denote the class of entire functions \(g(z_1,\ldots,z_n)\) of finite degree \((\nu_1,\ldots,\nu_n)\) that belong to the space \(\Lambda_{P,\varphi}^{(n)}\). Obviously, the class \(W_{\nu_1,\ldots,\nu_n}^{(P,\varphi)}\) in the case \(\varphi\equiv 1\) and \(p_1=p_2=\cdots=p_n=p\) coincides with the class \(W_{\nu_1,\ldots,\nu_n}^{(n)}\).

In the case when \(p_1,p_2,\ldots,p_n\) are various numbers not less than unity, and \(\varphi(x_1,\ldots,x_n)\equiv 1\), the notations used are

\[ W_{\nu_1,\ldots,\nu_n}^{(p,1)} \equiv W_{\nu_1,\ldots,\nu_n}^{(p_1,\ldots,p_n)}, \qquad \|f\|_{P,1}^{(n)}=\|f\|_{p_1,\ldots,p_n}. \]

The second problem* of the present note consists in establishing a connection between the different norms \(\|g\|_{p_1,\ldots,p_n}\) and \(\|g\|_{p'_1,\ldots,p'_n}\) of an entire function \(g(z_1,\ldots,z_n)\) from the class \(W_{\nu_1,\ldots,\nu_n}^{(p_1,\ldots,p_n)}\), where \(1\leqslant p_i<p'_i\leqslant\infty\) \((i=1,2,\ldots,n)\).

\[ \text{*} \]

• A less precise result with respect to the constant was obtained by the author, by another method, in the work \((^6)\), carried out jointly with A. S. Dzhafarov, where a connection was established between the different norms \(\|g\|_{P',\varphi}^{(n)}\) and \(\|g\|_{P,\varphi}^{(n)}\) in the class \(W_{\nu_1,\ldots,\nu_n}^{(P,\varphi)}\), with \(P=(p_1,\ldots,p_n)\), \(P'=(p'_1,\ldots,p'_n)\), and \(1\leqslant p_i<p'_i\leqslant\infty\) \((i=1,2,\ldots,n)\).

1. If \(g(z_1,\ldots,z_n)\in W_{\nu_1,\ldots,\nu_n}^{(p_1,\ldots,p_n)}\) and \(p_1,p_2,\ldots,p_n\) are distinct numbers not less than one, then

\[ \max_{-\infty<x_1,\ldots,x_n<\infty}|g(x_1,\ldots,x_n)| \le \prod_{k=1}^{n}\left(\frac{s_k\nu_k}{\pi}\right)^{1/p_k} \|g\|_{p_1,\ldots,p_n}, \]

where \(s_k=|[-p_k/2]|\) is the least integer not less than \(p_k/2\) \((k=1,2,\ldots,n)\).

1. If \(g(z_1,\ldots,z_n)\in W_{\nu_1,\ldots,\nu_n}^{(p_1,\ldots,p_n)}\); \(p_1,p_2,\ldots,p_n,\ p'_1,p'_2,\ldots,p'_n\) are distinct numbers not less than one, and \(1\le p_i\le p'_i\le\infty\) \((i=1,2,\ldots,n)\), then we have*

\[ \|g\|_{p'_1,\ldots,p'_n} \le \prod_{k=1}^{n}\left(\frac{s_k\nu_k}{\pi}\right)^{1/p_k-1/p'_k} \|g\|_{p_1,\ldots,p_n}, \tag{7} \]

where

\[ s_k=|[-p_k/2]| \quad (k=1,2,\ldots,n). \]

Institute of Mathematics and Mechanics
Academy of Sciences of the Azerbaijan SSR

Received
3 V 1963

REFERENCES

¹ I. I. Ibragimov, Extremal Properties of Entire Functions of Finite Degree, Baku, 1962.
² E. Stein, Ann. Math., 65, No. 3 (1957).
³ S. M. Nikol’skii, Tr. Mat. Inst. im. V. A. Steklova AN SSSR, 38 (1951).
⁴ I. I. Ibragimov, Izv. AN SSSR, ser. matem., 23, 243 (1959).
⁵ I. I. Ibragimov, A. S. Dzhafarov, DAN, 138, No. 4 (1961).
⁶ I. I. Ibragimov, A. S. Dzhafarov, Izv. AN AzerbSSR, ser. phys.-math. and techn. sciences, No. 5 (1962).
⁷ S. M. Nikol’skii, Siberian Math. Journal, 3, No. 6 (1962).

* A special case of inequality (7) with a nonsharp constant was considered independently of us by S. M. Nikol’skii (⁷).